A minimal mathematical example of polycomputing

Polycomputing is the ability of a material to provide the results of multiple computations in the same place at the same time. A minimal mathematical example of polycomputing is a one-element partial order with a reflexive relation denoting “less than or equal to”. Let A be the set equipped with the order and a be the sole element of A. Then a is simultaneously the maximum element of A, being greater than or equal to every other element in A (i.e., to itself), and it is the minimum element of A, being less than or equal to every other element in A. Similarly, a monotonic function from A to itself is simultaneously increasing and decreasing, and simultaneously order-preserving and order-reversing.

The computations a material (which in the mathematical case is apparently a system of relations such as a partial order) can perform are observed differently by different observers. To observe polycomputing, we have to be able to take multiple perspectives on a material. In a partial order, the same element in a given web of relationships can be seen as a maximum element, a minimum element, or both, depending on the observer perspective we take, as I show in pages 10-11 of my essay on memory in math. Another example of mathematical polycomputing is sets and functions. If you think of functions as the perspectives that sets take on each other, then the output of a function f given an input x yields different results depending on what f is, which sounds like polycomputing. Or, for a given function, the computation it outputs depends on the varying input x, which again could be thought of as polycomputing. In a group such as the integers equipped with addition, the result of a computation such as 2 + z depends on what z is. This is polycomputing if you accept that every 2 + z (i.e., 2 + z for all integers z) are computed in the same place (i.e., in the group) and at the same time (because math doesn’t have time). This may seem trivial, but I think it’s noteworthy that some ideas that are apparently radical in the world of science are apparently obvious in the world of math.

I share other examples of polycomputing in my essay on relational realism, such as the ability of a tomato to be a fruit and a vegetable simultaneously. (Relational realism is probably a generalization of polycomputing that drops the “computing” concept.) Lisa Feldman Barrett calls the process of changing the observed output of a computation depending on the observer perspective taken “meaning-making”. Here is another example of polycomputing in the real world: A stop sign approached from one position on the road means “stop”; the same stop sign when seen from another perspective means “it is okay to go (because the driver that would otherwise hit you has to stop)”. In the Esther Thelen view of infant locomotion, a leg motion that might be seen as “kicking” from an isolated focus on the individual leg might instead be seen as “crawling” when observed in relation to the rest of what the body is doing.

Prices are another example of polycomputing. Prices acquire meaning in relation to other prices. For example, if an apple costs $1 and an orange costs $2, then the $1 price of apples means “half an orange” (as in, buying one apple means giving up half an orange that you can no longer buy). As a result, the same price means different things depending on the comparison made; a given item of produce for sale in a space such as a grocery store at a given moment in time yields many different computational results depending on the perspectie taken. This is very powerful because it lets shoppers choose the perspectives that matter to them while ignoring all others. It also allows new goods to enter the system and goods that are no longer worth producing to exit the system because the polycomputational system updates naturally and automatically as new comparisons enter and exit.

The fact that both stop signs and prices are examples of polycomputing and are also examples of social coordination technologies suggests that polycomputing is crucial to coordination. This may be because polycomputing allows two agents to see the same object from different perspectives and know that while they are receiving one signal, the other agent is receiving a different signal such that their plans based on those signals are likely to be mutually compatible.

Polycomputing is an important source of variation. Variation is crucial to development because it allows the system to discover, test, and explore new possibilities. For example, an infant can discover that the same bed that is good for sleeping in is also good for bouncing in, leading to the development of new games.